Problem 30 - Precalculus Mathematics for Calculus

Question

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\). $$\left(-\frac{3}{5}, \frac{4}{5}\right)$$

Step-by-Step Solution

Answer

\(\sin t = \frac{4}{5}\), \(\cos t = -\frac{3}{5}\), \(\tan t = -\frac{4}{3}\).

Step 1: Identify the coordinates

The terminal point is given as \(P\left(-\frac{3}{5}, \frac{4}{5}\right)\). Here, \(x = -\frac{3}{5}\) and \(y = \frac{4}{5}\).

Step 2: Calculate \(\sin t\)

In a unit circle, \(\sin t\) is the y-coordinate of the terminal point. This means \(\sin t = \frac{4}{5}\).

Step 3: Calculate \(\cos t\)

In a unit circle, \(\cos t\) is the x-coordinate of the terminal point. This means \(\cos t = -\frac{3}{5}\).

Step 4: Calculate \(\tan t\)

The tangent function is the ratio of the sine to the cosine. Therefore, \(\tan t = \frac{\sin t}{\cos t} = \frac{\frac{4}{5}}{-\frac{3}{5}} = -\frac{4}{3}\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. This circle allows us to define the trigonometric functions for all angles. When we say unit circle, we often refer to the standard unit circle where:

• The center is at the origin (0,0)
• The radius is exactly 1
• The circumference interacts with the x and y axes

Each point on the unit circle corresponds to a real number, often denoted as the angle \( t \). The coordinates of this point \( (x, y) \) tell us about specific trigonometric values. By traveling around the unit circle, the x and y values help us understand the sine, cosine, and tangent functions deeply. This is because on a unit circle, these coordinates are proportional to those trigonometric functions.

Sine Function

The sine function is one of the primary trigonometric functions and is often abbreviated as \( \sin \). On the unit circle, the sine of an angle \( t \) is represented by the y-coordinate of the terminal point. This shows \( \sin t \) as the vertical projection of the radius line that forms angle \( t \) with the positive x-axis.When you see an angle drawn on the unit circle, the sine function tells you how high the terminal point is from the horizontal axis. For example, if the terminal point \( P(x, y) \) is \( \left(-\frac{3}{5}, \frac{4}{5}\right) \), then the sine value is \( \sin t = \frac{4}{5} \). The sine function helps determine the vertical component for any angle placed on the circle.

Cosine Function

The cosine function, noted as \( \cos \), connects strongly with the unit circle as well. On the unit circle, cosine of an angle \( t \) can be defined using the x-coordinate of the terminal point associated with that angle. Simply put, \( \cos t \) describes the horizontal distance of the point on the circle.So, when you talk about the angle with reference to the x-axis, cosine helps measure how far along the x-axis the point is. If the coordinates are \( \left(-\frac{3}{5}, \frac{4}{5}\right) \), then \( \cos t = -\frac{3}{5} \). Cosine is especially useful in physics and engineering, when you need to know the horizontal reach or extension due to an angle.

Tangent Function

The tangent function is a little distinct compared to sine and cosine, yet closely related. It is usually written as \( \tan \). This function is the ratio of the y-coordinate (sine) to the x-coordinate (cosine) on the unit circle.

• So, \( \tan t = \frac{\sin t}{\cos t} \)
• For the point \( \left(-\frac{3}{5}, \frac{4}{5}\right) \), it becomes \( \tan t = \frac{\frac{4}{5}}{-\frac{3}{5}} = -\frac{4}{3} \)

The tangent gives you a hint of direction or slope, classically in relation to the angle and line created by moving from the origin to the point \( P(x, y) \) on the unit circle. It can help describe the steepness of a line in graphical and geometric representations.